I stumbled upon something that confused me recently regarding topics in differential geometry.
From what I understand, one of the key parts of the definition of a manifold is that it is locally flat or isomorphic to Euclidean space.
However, does this mean that locally, it's metric is also Euclidean?
The reason I ask is because what if you have a manifold with intrinsic curvature, say a 2 sphere. A 2 sphere is locally flat but I also know that it's metric is not Euclidean.
Can someone clarify for me?
Perhaps the mistake I am making is that they have nothing to do with one another, and that being locally isomorphic to Euclidean space doesn't imply a Euclidean metric.
"Locally flat" means the metric (which is an additional structure on top of a manifold's differential structure) has 0 curvature.
Any manifold is "locally Euclidean" meaning that it is locally homeomorphic to $\mathbb R^n$. This is what I assume you're referring to in the manifold definition. This does not refer to any metric structure.
The standard 2-sphere is thought of as a submanifold of $\mathbb R^3$, from which it is assumed to inherit a standard metric. With this structure it becomes a Riemannian manifold, whose curvature you can compute (it is not locally flat).
I hope that helps explain some of the terminology.